To solve for \( x \) in the equation
\[ \frac{8}{11} \left( x + \frac{4}{5} \right) = \frac{24}{55}, \]
we first isolate the term inside the parentheses. We can do this by multiplying both sides of the equation by the reciprocal of \(\frac{8}{11}\), which is \(\frac{11}{8}\):
\[ x + \frac{4}{5} = \frac{24}{55} \cdot \frac{11}{8}. \]
Calculating the right-hand side:
\[ \frac{24 \cdot 11}{55 \cdot 8} = \frac{264}{440}. \]
Now, simplify \(\frac{264}{440}\):
The GCD of 264 and 440 is 88 (by factoring), \[ 264 \div 88 = 3, \quad 440 \div 88 = 5. \] Thus, \[ \frac{264}{440} = \frac{3}{5}. \]
Now, we have:
\[ x + \frac{4}{5} = \frac{3}{5}. \]
Next, we isolate \( x \) by subtracting \(\frac{4}{5}\) from both sides:
\[ x = \frac{3}{5} - \frac{4}{5}. \]
Calculating this gives:
\[ x = \frac{3 - 4}{5} = \frac{-1}{5}. \]
Thus, the solution for \( x \) is
\[ \boxed{-\frac{1}{5}}. \]